The supertail of a subspace partition

Let V = V(n, q) be a vector space of dimension n over the finite field with q elements, and let d (1) < d (2) < ... < d (m) be the dimensions that occur in a subspace partition of V. Let sigma (q) (n, t) denote the minimum size of a subspace partition of V, in which t is the largest dimension of a subspace. For any integer s, with 1 < s a parts per thousand currency sign m, the set of subspaces in of dimension less than d (s) is called the s-supertail of . The main result is that the number of spaces in an s-supertail is at least sigma (q) (d (s) , d (s-1)).